What is the Wedge Product and How is it Defined?

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Discussion Overview

The discussion revolves around the definition and properties of the wedge product in the context of vector spaces and exterior algebra. Participants express confusion regarding the associativity of the wedge product and its implications for expressions involving multiple vectors.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants express confusion about the definition of the wedge product as u \wedge v = u \otimes v - v \otimes u, questioning its associativity.
  • Others note that the problem does not attempt to define an associative algebra, specifically the exterior algebra, despite providing definitions for multiple wedge products.
  • It is suggested that if the wedge product is not associative, then expressions like u \wedge v \wedge w lack a defined order of operations.
  • One participant proposes considering the wedge product as a single symbol rather than a product of two wedges, suggesting a method for defining the operation based on permutations of the symbols involved.

Areas of Agreement / Disagreement

Participants generally agree on the confusion surrounding the associativity of the wedge product and its implications for multi-vector expressions. However, there is no consensus on how to interpret or resolve these issues.

Contextual Notes

Limitations include the lack of clarity on the definitions provided for the wedge product and the implications of non-associativity on multi-vector expressions.

jdstokes
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I'm confused about the wording of this problem,

They define the wedge product by [itex]u \wedge v = u\otimes v - v\otimes u[/itex] but as far as I can tell this operation is not associative!
 
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jdstokes said:
I'm confused about the wording of this problem,

They define the wedge product by [itex]u \wedge v = u\otimes v - v\otimes u[/itex] but as far as I can tell this operation is not associative!

They have not tried to define an associative algebra, i.e., the exterior algebra, in this question.

They have defined meanings for the the abstract symbols [itex]u \wedge v[/itex], [itex]u \wedge v \wedge w[/itex], and [itex]u \wedge v \wedge w \wedge x[/itex], where [itex]u[/itex], [itex]v[/itex], [itex]w[/itex], and [itex]x[/itex] are all arbitrary vectors.

Their definitions can be extended to the definition of the exterior algebra, but this is not needed to do the question.
 
If the operation is not associative then an expression such as [itex]u \wedge v \wedge w[/itex] is not defined because the order of operations is not specified.
 
jdstokes said:
If the operation is not associative then an expression such as [itex]u \wedge v \wedge w[/itex] is not defined because the order of operations is not specified.

They have defined what this symbol means. Don't think of it as the product of two wedges, think of it as one symbol.

Consider all possible permutations of the juxtaposed symbols [itex]uvw[/itex]. Stick a + in front of the permutation if it is an even permutation of [itex]uvw[/itex] and and a - in front if the permutation is odd. Add all the terms, and insert tensor product symbols.
 

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